The Jarzynski equality is one of the most influential results in the field of non equilibrium statistical mechanics. This celebrated equality allows to calculate equilibrium free energy differences from work distributions of nonequilibrium processes. In practice, such calculations often suffer from poor convergence experimentally as well as numerically due to the need to sample rare events. Here we examine if the inclusion of a Maxwellian demon - measurement and feedback - can reduce the number nonequilibrium trials required for the estimation of the free energy. A modified version of the Jarzynski equality in which realizations with a given outcome are kept, while others are discarded, is used. We find that discarding realizations with unwanted outcomes can result in less trials compared to calculations based on the Jarzynski equality. We argue that this is closely related to Bennett's Acceptance Ratio (BAR), which was developed without any reference to measurements or feedback.